Poincar\'e series of double coset representatives of Coxeter groups

TL;DR

This paper investigates the rationality of Poincaré series for double coset representatives in Coxeter groups, establishing conditions related to normalizers of finite parabolic subgroups and proving rationality for affine Weyl groups.

Contribution

It links the rationality of double coset Poincaré series to that of normalizers of finite parabolic subgroups and proves rationality for affine Weyl groups.

Findings

Poincaré series rationality depends on normalizers of finite parabolic subgroups.

All Poincaré series for affine Weyl groups are rational.

Explicit examples of these series are provided.

Abstract

Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We study the rationality of the Poincar\'e series of the set of representatives of minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends mostly on the rationality of the Poincar\'e series of the normalizers of finite parabolic subgroups of $W$. For affine Weyl groups, we prove that all these series are rational and we give some explicit examples.